
Physica D: Nonlinear Phenomena [SJR: 1.049] [HI: 102] [3 followers] Follow Hybrid journal (It can contain Open Access articles) ISSN (Print) 01672789 Published by Elsevier [3177 journals] 
 Gradual multifractal reconstruction of timeseries: Formulation of the
method and an application to the coupling between stock market indices and
their Hölder exponents Authors: Christopher J. Keylock
Pages: 1  9
Abstract: Publication date: 1 April 2018
Source:Physica D: Nonlinear Phenomena, Volume 368
Author(s): Christopher J. Keylock
A technique termed gradual multifractal reconstruction (GMR) is formulated. A continuum is defined from a signal that preserves the pointwise Hölder exponent (multifractal) structure of a signal but randomises the locations of the original data values with respect to this ( φ = 0 ), to the original signal itself( φ = 1 ). We demonstrate that this continuum may be populated with synthetic time series by undertaking selective randomisation of wavelet phases using a dualtree complex wavelet transform. That is, the φ = 0 end of the continuum is realised using the recently proposed iterated, amplitude adjusted wavelet transform algorithm (Keylock, 2017) that fully randomises the wavelet phases. This is extended to the GMR formulation by selective phase randomisation depending on whether or not the wavelet coefficient amplitudes exceeds a threshold criterion. An econophysics application of the technique is presented. The relation between the normalised logreturns and their Hölder exponents for the daily returns of eight financial indices are compared. One particularly noticeable result is the change for the two American indices (NASDAQ 100 and S&P 500) from a nonsignificant to a strongly significant (as determined using GMR) crosscorrelation between the returns and their Hölder exponents from before the 2008 crash to afterwards. This is also reflected in the skewness of the phase difference distributions, which exhibit a geographical structure, with Asian markets not exhibiting significant skewness in contrast to those from elsewhere globally.
PubDate: 20180226T14:48:37Z
DOI: 10.1016/j.physd.2017.11.011
Issue No: Vol. 368 (2018)
 Authors: Christopher J. Keylock
 Synchronisation under shocks: The Lévy Kuramoto model
 Authors: Dale Roberts; Alexander C. Kalloniatis
Pages: 10  21
Abstract: Publication date: 1 April 2018
Source:Physica D: Nonlinear Phenomena, Volume 368
Author(s): Dale Roberts, Alexander C. Kalloniatis
We study the Kuramoto model of identical oscillators on Erdős–Rényi (ER) and Barabasi–Alberts (BA) scale free networks examining the dynamics when perturbed by a Lévy noise. Lévy noise exhibits heavier tails than Gaussian while allowing for their tempering in a controlled manner. This allows us to understand how ‘shocks’ influence individual oscillator and collective system behaviour of a paradigmatic complex system. Skewed α stable Lévy noise, equivalent to fractional diffusion perturbations, are considered, but overlaid by exponential tempering of rate λ . In an earlier paper we found that synchrony takes a variety of forms for identical Kuramoto oscillators subject to stable Lévy noise, not seen for the Gaussian case, and changing with α : a noiseinduced drift, a smooth α dependence of the point of crossover of synchronisation point of ER and BA networks, and a severe loss of synchronisation at low values of α . In the presence of tempering we observe both analytically and numerically a dramatic change to the α < 1 behaviour where synchronisation is sustained over a larger range of values of the ‘noise strength’ σ , improved compared to the α > 1 tempered cases. Analytically we study the system close to the phase synchronised fixed point and solve the tempered fractional Fokker–Planck equation. There we observe that densities show stronger support in the basin of attraction at low α for fixed coupling, σ and tempering λ . We then perform numerical simulations for networks of size N = 1000 and average degree d ̄ = 10 . There, we compute the order parameter r as a function of σ for fixed α and λ and observe values of r ≈ 1 over larger ranges of σ for α < 1 and λ ≠ 0 . In addition we observe drift of both positive and negative slopes for different α and λ when native frequencies are equal, and confirm a sustainment of synchronisation down to low values of α . We propose a mechanism for this in terms of the basic shape of the tempered stable Lévy densities for various α and how it feeds into Kuramoto oscillator dynamics and illustrate this with examples of specific paths.
PubDate: 20180226T14:48:37Z
DOI: 10.1016/j.physd.2017.12.005
Issue No: Vol. 368 (2018)
 Authors: Dale Roberts; Alexander C. Kalloniatis
 KdV equation beyond standard assumptions on initial data
 Authors: Alexei Rybkin
Pages: 1  11
Abstract: Publication date: 15 February 2018
Source:Physica D: Nonlinear Phenomena, Volume 365
Author(s): Alexei Rybkin
We show that the Cauchy problem for the KdV equation can be solved by the inverse scattering transform (IST) for any initial data bounded from below, decaying sufficiently rapidly at + ∞ , but unrestricted otherwise. Thus our approach does not require any boundary condition at − ∞ .
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2017.10.005
Issue No: Vol. 365 (2018)
 Authors: Alexei Rybkin
 Kink dynamics in a system of two coupled scalar fields in two
space–time dimensions Authors: A. AlonsoIzquierdo
Pages: 12  26
Abstract: Publication date: 15 February 2018
Source:Physica D: Nonlinear Phenomena, Volume 365
Author(s): A. AlonsoIzquierdo
In this paper we examine the scattering processes among the members of a rich family of kinks which arise in a (1+1)dimensional relativistic two scalar field theory. These kinks carry two different topological charges that determine the mutual interactions between the basic energy lumps (extended particles) described by these topological defects. Processes like topological charge exchange, kink–antikink bound state formation or kink repulsion emerge depending on the charges of the scattered particles. Twobounce resonant windows have been found in the antikink–kink scattering processes, but not in the kink–antikink interactions.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2017.10.006
Issue No: Vol. 365 (2018)
 Authors: A. AlonsoIzquierdo
 Homogenization of onedimensional draining through heterogeneous porous
media including higherorder approximations Authors: Daniel M. Anderson; Richard M. McLaughlin; Cass T. Miller
Pages: 42  56
Abstract: Publication date: 15 February 2018
Source:Physica D: Nonlinear Phenomena, Volume 365
Author(s): Daniel M. Anderson, Richard M. McLaughlin, Cass T. Miller
We examine a mathematical model of onedimensional draining of a fluid through a periodicallylayered porous medium. A porous medium, initially saturated with a fluid of a high density is assumed to drain out the bottom of the porous medium with a second lighter fluid replacing the draining fluid. We assume that the draining layer is sufficiently dense that the dynamics of the lighter fluid can be neglected with respect to the dynamics of the heavier draining fluid and that the height of the draining fluid, represented as a free boundary in the model, evolves in time. In this context, we neglect interfacial tension effects at the boundary between the two fluids. We show that this problem admits an exact solution. Our primary objective is to develop a homogenization theory in which we find not only leadingorder, or effective, trends but also capture higherorder corrections to these effective draining rates. The approximate solution obtained by this homogenization theory is compared to the exact solution for two cases: (1) the permeability of the porous medium varies smoothly but rapidly and (2) the permeability varies as a piecewise constant function representing discrete layers of alternating high/low permeability. In both cases we are able to show that the corrections in the homogenization theory accurately predict the position of the free boundary moving through the porous medium.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2017.10.010
Issue No: Vol. 365 (2018)
 Authors: Daniel M. Anderson; Richard M. McLaughlin; Cass T. Miller
 Patterns and coherence resonance in the stochastic Swift–Hohenberg
equation with Pyragas control: The Turing bifurcation case Authors: R. Kuske; C.Y. Lee; V. Rottschäfer
Pages: 57  71
Abstract: Publication date: 15 February 2018
Source:Physica D: Nonlinear Phenomena, Volume 365
Author(s): R. Kuske, C.Y. Lee, V. Rottschäfer
We provide a multiple time scales analysis for the Swift–Hohenberg equation with delayed feedback via Pyragas control, with and without additive noise. An analysis of the pattern formation near onset indicates both the possibility of either standing waves (rolls) or traveling waves via Turing or Turing–Hopf bifurcations, respectively, depending on the product of the strength of the feedback and the length of the delay. The remainder of the paper is focused on Turing bifurcations, where the delay can drive the appearance of an additional time scale, intermediate to the usual slow and fast time scales observed in the modulation of rolls without delay. In the deterministic case, a Ginzburg–Landautype modulation equation is derived that inherits Pyragas control terms from the original equation. The Eckhaus stability criteria is obtained for the rolls, with the intermediate time scale observed in the transients. In the stochastic context, slow modulation equations are derived for the amplitudes of the primary modes that are coupled to a fast Ornstein–Uhlenbecktype equation with delay for the zero mode driven by the additive noise. By deriving an averaging approximation for the amplitude of the primary mode, we show how the interaction of noise and delay influences the existence and stability range for the noisy rolltype patterns. Furthermore, approximations for the spectral densities of the primary and zero modes show that oscillations on the intermediate times scale are sustained through the phenomenon of coherence resonance. These dynamics on the intermediate time scale are sustained through the interaction of noise and delay, in contrast to the deterministic context where dynamics on the intermediate times scale are transient.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2017.10.012
Issue No: Vol. 365 (2018)
 Authors: R. Kuske; C.Y. Lee; V. Rottschäfer
 Oscillating solutions of the Vlasov–Poisson system—A numerical
investigation Authors: Tobias Ramming; Gerhard Rein
Pages: 72  79
Abstract: Publication date: 15 February 2018
Source:Physica D: Nonlinear Phenomena, Volume 365
Author(s): Tobias Ramming, Gerhard Rein
Numerical evidence is given that spherically symmetric perturbations of stable spherically symmetric steady states of the gravitational Vlasov–Poisson system lead to solutions which oscillate in time. The oscillations can be periodic in time or damped. Along oneparameter families of polytropic steady states we establish an Eddington–Ritter type relation which relates the period of the oscillation to the central density of the steady state. The numerically obtained periods are used to estimate possible periods for typical elliptical galaxies.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2017.10.013
Issue No: Vol. 365 (2018)
 Authors: Tobias Ramming; Gerhard Rein
 Turbulence in vertically averaged convection
 Authors: N. Balci; A.M. Isenberg; M.S. Jolly
Abstract: Publication date: Available online 23 February 2018
Source:Physica D: Nonlinear Phenomena
Author(s): N. Balci, A.M. Isenberg, M.S. Jolly
The vertically averaged velocity of the 3D RayleighBénard problem is analyzed and numerically simulated. This vertically averaged velocity satisfies a 2D incompressible Navier–Stokes system with a body force involving the 3D velocity. A time average of this force is estimated through time averages of the 3D velocity. Relations similar to those from 2D turbulence are then derived. Direct numerical simulation of the 3D Rayleigh Bénard are carried out to test how prominent the features of 2D turbulence are for this Navier–Stokes system.
PubDate: 20180226T14:48:37Z
DOI: 10.1016/j.physd.2018.02.005
 Authors: N. Balci; A.M. Isenberg; M.S. Jolly
 Wave breaking for the Stochastic Camassa–Holm equation
 Authors: Dan Crisan; Darryl D. Holm
Abstract: Publication date: Available online 16 February 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Dan Crisan, Darryl D. Holm
We show that wave breaking occurs with positive probability for the Stochastic Camassa–Holm (SCH) equation. This means that temporal stochasticity in the diffeomorphic flow map for SCH does not prevent the wave breaking process which leads to the formation of peakon solutions. We conjecture that the timeasymptotic solutions of SCH will consist of emergent wave trains of peakons moving along stochastic space–time paths.
PubDate: 20180226T14:48:37Z
DOI: 10.1016/j.physd.2018.02.004
 Authors: Dan Crisan; Darryl D. Holm
 The stability and slow dynamics of spot patterns in the 2D Brusselator
model: The effect of open systems and heterogeneities Authors: J.C. Tzou; M.J. Ward
Abstract: Publication date: Available online 13 February 2018
Source:Physica D: Nonlinear Phenomena
Author(s): J.C. Tzou, M.J. Ward
Spot patterns, whereby the activator field becomes spatially localized near certain dynamicallyevolving discrete spatial locations in a bounded multidimensional domain, is a common occurrence for twocomponent reaction–diffusion (RD) systems in the singular limit of a large diffusivity ratio. In previous studies of 2D localized spot patterns for various specific wellknown RD systems, the domain boundary was assumed to be impermeable to both the activator and inhibitor, and the reactionkinetics were assumed to be spatially uniform. As an extension of this previous theory, we use formal asymptotic methods to study the existence, stability, and slow dynamics of localized spot patterns for the singularly perturbed 2D Brusselator RD model when the domain boundary is only partially impermeable, as modeled by an inhomogeneous Robin boundary condition, or when there is an influx of inhibitor across the domain boundary. In our analysis, we will also allow for the effect of a spatially variable bulk feed term in the reaction kinetics. By applying our extended theory to the special case of onespot patterns and ring patterns of spots inside the unit disk, we provide a detailed analysis of the effect on spot patterns of these three different sources of heterogeneity. In particular, when there is an influx of inhibitor across the boundary of the unit disk, a ring pattern of spots can become pinned to a ringradius closer to the domain boundary. Under a Robin condition, a quasiequilibrium ring pattern of spots is shown to exhibit a novel saddle–node bifurcation behavior in terms of either the inhibitor diffusivity, the Robin constant, or the ambient background concentration. A spatially variable bulk feed term, with a concentrated source of “fuel” inside the domain, is shown to yield a saddle–node bifurcation structure of spot equilibria, which leads to qualitatively new spotpinning behavior. Results from our asymptotic theory are validated from full numerical simulations of the Brusselator model.
PubDate: 20180226T14:48:37Z
DOI: 10.1016/j.physd.2018.02.002
 Authors: J.C. Tzou; M.J. Ward
 Krein signature for instability of PTsymmetric states
 Authors: Alexander Chernyavsky; Dmitry E. Pelinovsky
Abstract: Publication date: Available online 9 February 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Alexander Chernyavsky, Dmitry E. Pelinovsky
Krein quantity is introduced for isolated neutrally stable eigenvalues associated with the stationary states in the PT symmetric nonlinear Schrödinger equation. Krein quantity is real and nonzero for simple eigenvalues but it vanishes if two simple eigenvalues coalesce into a defective eigenvalue. A necessary condition for bifurcation of unstable eigenvalues from the defective eigenvalue is proved. This condition requires the two simple eigenvalues before the coalescence point to have opposite Krein signatures. The theory is illustrated with several numerical examples motivated by recent publications in physics literature.
PubDate: 20180226T14:48:37Z
DOI: 10.1016/j.physd.2018.01.009
 Authors: Alexander Chernyavsky; Dmitry E. Pelinovsky
 The inviscid limit of the incompressible 3D Navier–Stokes equations
with helical symmetry Authors: Quansen Jiu; Milton C. Lopes Filho; Dongjuan Niu; Helena J. Nussenzveig Lopes
Abstract: Publication date: Available online 5 February 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Quansen Jiu, Milton C. Lopes Filho, Dongjuan Niu, Helena J. Nussenzveig Lopes
In this paper, we are concerned with the vanishing viscosity problem for the threedimensional Navier–Stokes equations with helical symmetry, in the whole space. We choose viscositydependent initial u 0 ν with helical swirl, an analogue of the swirl component of axisymmetric flow, of magnitude O ( ν ) in the L 2 norm; we assume u 0 ν → u 0 in H 1 . The new ingredient in our analysis is a decomposition of helical vector fields, through which we obtain the required estimates.
PubDate: 20180226T14:48:37Z
DOI: 10.1016/j.physd.2018.01.013
 Authors: Quansen Jiu; Milton C. Lopes Filho; Dongjuan Niu; Helena J. Nussenzveig Lopes
 Scalefree behavior of networks with the copresence of preferential and
uniform attachment rules Authors: Angelica Pachon; Laura Sacerdote; Shuyi Yang
Abstract: Publication date: Available online 3 February 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Angelica Pachon, Laura Sacerdote, Shuyi Yang
Complex networks in different areas exhibit degree distributions with a heavy upper tail. A preferential attachment mechanism in a growth process produces a graph with this feature. We herein investigate a variant of the simple preferential attachment model, whose modifications are interesting for two main reasons: to analyze more realistic models and to study the robustness of the scalefree behavior of the degree distribution. We introduce and study a model which takes into account two different attachment rules: a preferential attachment mechanism (with probability 1 − p ) that stresses the rich get richer system, and a uniform choice (with probability p ) for the most recent nodes, i.e. the nodes belonging to a window of size w to the left of the last born node. The latter highlights a trend to select one of the last added nodes when no information is available. The recent nodes can be either a given fixed number or a proportion ( α n ) of the total number of existing nodes. In the first case, we prove that this model exhibits an asymptotically powerlaw degree distribution. The same result is then illustrated through simulations in the second case. When the window of recent nodes has a constant size, we herein prove that the presence of the uniform rule delays the starting time from which the asymptotic regime starts to hold. The mean number of nodes of degree k and the asymptotic degree distribution are also determined analytically. Finally, a sensitivity analysis on the parameters of the model is performed.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2018.01.005
 Authors: Angelica Pachon; Laura Sacerdote; Shuyi Yang
 A thirdorder classD amplifier with and without ripple compensation
 Authors: Stephen M. Cox; H. du Toit Mouton
Abstract: Publication date: Available online 3 February 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Stephen M. Cox, H. du Toit Mouton
We analyse the nonlinear behaviour of a thirdorder classD amplifier, and demonstrate the remarkable effectiveness of the recently introduced ripple compensation (RC) technique in reducing the audio distortion of the device. The amplifier converts an input audio signal to a highfrequency train of rectangular pulses, whose widths are modulated according to the input signal (pulsewidth modulation) and employs negative feedback. After determining the steadystate operating point for constant input and calculating its stability, we derive a smallsignal model (SSM), which yields in closed form the transfer function relating (infinitesimal) input and output disturbances. This SSM shows how the RC technique is able to linearise the smallsignal response of the device. We extend this SSM through a fully nonlinear perturbation calculation of the dynamics of the amplifier, based on the disparity in time scales between the pulse train and the audio signal. We obtain the nonlinear response of the amplifier to a general audio signal, avoiding the linearisation inherent in the SSM; we thereby more precisely quantify the reduction in distortion achieved through RC. Finally, simulations corroborate our theoretical predictions and illustrate the dramatic deterioration in performance that occurs when the amplifier is operated in an unstable regime. The perturbation calculation is rather general, and may be adapted to quantify the way in which other nonlinear negativefeedback pulsemodulated devices track a timevarying input signal that slowly modulates the system parameters.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2018.01.012
 Authors: Stephen M. Cox; H. du Toit Mouton
 Pseudosimple heteroclinic cycles in R4
 Authors: Pascal Chossat; Alexander Lohse; Olga Podvigina
Abstract: Publication date: Available online 2 February 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Pascal Chossat, Alexander Lohse, Olga Podvigina
We study pseudosimple heteroclinic cycles for a Γ equivariant system in R 4 with finite Γ ⊂ O ( 4 ) , and their nearby dynamics. In particular, in a first step towards a full classification –analogous to that which exists already for the class of simple cycles –we identify all finite subgroups of O ( 4 ) admitting pseudosimple cycles. To this end we introduce a constructive method to build equivariant dynamical systems possessing a robust heteroclinic cycle. Extending a previous study we also investigate the existence of periodic orbits close to a pseudosimple cycle, which depends on the symmetry groups of equilibria in the cycle. Moreover, we identify subgroups Γ ⊂ O ( 4 ) , Γ ⊄ S O ( 4 ) , admitting fragmentarily asymptotically stable pseudosimple heteroclinic cycles. (It has been previously shown that for Γ ⊂ S O ( 4 ) pseudosimple cycles generically are completely unstable.) Finally, we study a generalized heteroclinic cycle, which involves a pseudosimple cycle as a subset.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2018.01.008
 Authors: Pascal Chossat; Alexander Lohse; Olga Podvigina
 Exact closedform solutions of a fully nonlinear asymptotic twofluid
model Authors: Alexei F. Cheviakov
Abstract: Publication date: Available online 31 January 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Alexei F. Cheviakov
A fully nonlinear model of Choi and Camassa (1999) describing onedimensional incompressible dynamics of two nonmixing fluids in a horizontal channel, under a shallow water approximation, is considered. An equivalence transformation is presented, leading to a special dimensionless form of the system, involving a single dimensionless constant physical parameter, as opposed to five parameters present in the original model. A firstorder dimensionless ordinary differential equation describing traveling wave solutions is analyzed. Several multiparameter families of physically meaningful exact closedform solutions of the twofluid model are derived, corresponding to periodic, solitary, and kinktype bidirectional traveling waves; specific examples are given, and properties of the exact solutions are analyzed.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2018.01.001
 Authors: Alexei F. Cheviakov
 Limit cycles via higher order perturbations for some piecewise
differential systems Authors: Claudio A. Buzzi; Maurício Firmino Silva Lima; Joan Torregrosa
Abstract: Publication date: Available online 31 January 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Claudio A. Buzzi, Maurício Firmino Silva Lima, Joan Torregrosa
A classical perturbation problem is the polynomial perturbation of the harmonic oscillator, ( x ′ , y ′ ) = ( − y + ε f ( x , y , ε ) , x + ε g ( x , y , ε ) ) . In this paper we study the limit cycles that bifurcate from the period annulus via piecewise polynomial perturbations in two zones separated by a straight line. We prove that, for polynomial perturbations of degree n , no more than N n − 1 limit cycles appear up to a study of order N . We also show that this upper bound is reached for orders one and two. Moreover, we study this problem in some classes of piecewise Liénard differential systems providing better upper bounds for higher order perturbation in ε , showing also when they are reached. The Poincaré–Pontryagin–Melnikov theory is the main technique used to prove all the results.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2018.01.007
 Authors: Claudio A. Buzzi; Maurício Firmino Silva Lima; Joan Torregrosa
 Bifurcation analysis of eight coupled degenerate optical parametric
oscillators Authors: Daisuke Ito; Tetsushi Ueta; Kazuyuki Aihara
Abstract: Publication date: Available online 31 January 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Daisuke Ito, Tetsushi Ueta, Kazuyuki Aihara
A degenerate optical parametric oscillator (DOPO) network realized as a coherent Ising machine can be used to solve combinatorial optimization problems. Both theoretical and experimental investigations into the performance of DOPO networks have been presented previously. However a problem remains, namely that the dynamics of the DOPO network itself can lower the search success rates of globally optimal solutions for Ising problems. This paper shows that the problem is caused by pitchfork bifurcations due to the symmetry structure of coupled DOPOs. Some twoparameter bifurcation diagrams of equilibrium points express the performance deterioration. It is shown that the emergence of nonground states regarding local minima hampers the system from reaching the ground states corresponding to the global minimum. We then describe a parametric strategy for leading a system to the ground state by actively utilizing the bifurcation phenomena. By adjusting the parameters to break particular symmetry, we find appropriate parameter sets that allow the coherent Ising machine to obtain the globally optimal solution alone.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2018.01.010
 Authors: Daisuke Ito; Tetsushi Ueta; Kazuyuki Aihara
 Generalized Lagrangian coherent structures
 Authors: Sanjeeva Balasuriya; Nicholas T. Ouellette; Irina I. Rypina
Abstract: Publication date: Available online 31 January 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Sanjeeva Balasuriya, Nicholas T. Ouellette, Irina I. Rypina
The notion of a Lagrangian Coherent Structure (LCS) is by now well established as a way to capture transient coherent transport dynamics in unsteady and aperiodic fluid flows that are known over finite time. We show that the concept of an LCS can be generalized to capture coherence in other quantities of interest that are transported by, but not fully locked to, the fluid. Such quantities include those with dynamic, biological, chemical, or thermodynamic relevance, such as temperature, pollutant concentration, vorticity, kinetic energy, plankton density, and so on. We provide a conceptual framework for identifying the Generalized Lagrangian Coherent Structures (GLCSs) associated with such evolving quantities. We show how LCSs can be seen as a special case within this framework, and provide an overarching discussion of various methods for identifying LCSs. The utility of this more general viewpoint is highlighted through a variety of examples. We also show that although LCSs approximate GLCSs in certain limiting situations under restrictive assumptions on how the velocity field affects the additional quantities of interest, LCSs are not in general sufficient to describe their coherent transport.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2018.01.011
 Authors: Sanjeeva Balasuriya; Nicholas T. Ouellette; Irina I. Rypina
 Nonequilibrium diffusive gas dynamics: Poiseuille microflow
 Authors: Rafail V. Abramov; Jasmine T. Otto
Abstract: Publication date: Available online 16 January 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Rafail V. Abramov, Jasmine T. Otto
We test the recently developed hierarchy of diffusive moment closures for gas dynamics together with the nearwall viscosity scaling on the Poiseuille flow of argon and nitrogen in a one micrometer wide channel, and compare it against the corresponding Direct Simulation Monte Carlo computations. We find that the diffusive regularized Grad equations with viscosity scaling provide the most accurate approximation to the benchmark DSMC results. At the same time, the conventional Navier–Stokes equations without the nearwall viscosity scaling are found to be the least accurate among the tested closures.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2018.01.006
 Authors: Rafail V. Abramov; Jasmine T. Otto
 Single bumps in a 2population homogenized neuronal network model
 Authors: Karina Kolodina; Anna Oleynik; John Wyller
Abstract: Publication date: Available online 11 January 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Karina Kolodina, Anna Oleynik, John Wyller
We investigate existence and stability of single bumps in a homogenized 2population neural field model, when the firing rate functions are given by the Heaviside function. The model is derived by means of the twoscale convergence technique of Nguetseng in the case of periodic microvariation in the connectivity functions. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. The bump solutions are constructed by using a pinning function technique for the case where the solutions are independent of the local variable. In the weakly modulated case the generic picture consists of two bumps (one narrow and one broad bump) for each admissible set of threshold values for firing. In addition, a new threshold value regime for existence of bumps is detected. Beyond the weakly modulated regime the number of bumps depends sensitively on the degree of heterogeneity. For the latter case we present a configuration consisting of three coexisting bumps. The linear stability of the bumps is studied by means of the spectral properties of a Fredholm integral operator, block diagonalization of this operator and the Fourier decomposition method. In the weakly modulated regime, one of the bumps is unstable for all relative inhibition times, while the other one is stable for small and moderate values of this parameter. The latter bump becomes unstable as the relative inhibition time exceeds a certain threshold. In the case of the three coexisting bumps detected in the regime of finite degree of heterogeneity, we have at least one stable bump (and maximum two stable bumps) for small and moderate values of the relative inhibition time.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2018.01.004
 Authors: Karina Kolodina; Anna Oleynik; John Wyller
 Long time behavior of the twodimensional Boussinesq equations without
buoyancy diffusion Authors: Charles R. Doering; Jiahong Wu; Kun Zhao; Xiaoming Zheng
Abstract: Publication date: Available online 10 January 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Charles R. Doering, Jiahong Wu, Kun Zhao, Xiaoming Zheng
We study the global wellposedness and stability/instability of perturbations near a special type of hydrostatic equilibrium associated with the 2D Boussinesq equations without buoyancy (a.k.a. thermal) diffusion on a bounded domain subject to stressfree boundary conditions. The boundary of the domain is not necessarily smooth and may have corners such as in the case of rectangles. We achieve three goals. First, we establish the globalintime existence and uniqueness of largeamplitude classical solutions. Efforts are made to reduce the regularity assumptions on the initial data. Second, we obtain the largetime asymptotics of the full nonlinear perturbation. In particular, we show that the kinetic energy and the first order derivatives of the velocity field converge to zero as time goes to infinity, regardless of the magnitude of the initial data, and the flow stratifies in the vertical direction in a weak topology. Third, we prove the linear stability of the hydrostatic equilibrium T ( y ) satisfying T ′ ( y ) = α > 0 , and the linear instability of periodic perturbations when T ′ ( y ) = α < 0 . Numerical simulations are supplemented to corroborate the analytical results and predict some phenomena that are not proved. The authors are pleased to dedicate this paper to Professor Edriss Saleh Titi on the occasion of his 60th birthday. Professor Titi’s myriad research contributions – including contributions to the problem constituting the focus of this work – and leadership in the mathematical fluid dynamics community serve as an inspiration to his students, collaborators and colleagues.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2017.12.013
 Authors: Charles R. Doering; Jiahong Wu; Kun Zhao; Xiaoming Zheng
 Traveling waves in a spatiallydistributed Wilson–Cowan model of
cortex: From fronts to pulses Authors: Jeremy D. Harris; Bard Ermentrout
Abstract: Publication date: Available online 2 January 2018
Source:Physica D: Nonlinear Phenomena
Author(s): Jeremy D. Harris, Bard Ermentrout
Wave propagation in excitable media has been studied in various biological, chemical, and physical systems. Waves are among the most common evoked and spontaneous organized activity seen in cortical networks. In this paper, we study traveling fronts and pulses in a spatiallyextended version of the Wilson–Cowan equations, a neural firing rate model of sensory cortex having two population types: Excitatory and inhibitory. We are primarily interested in the case when the local or spaceclamped dynamics has three fixed points: (1) a stable down state; (2) a saddle point with stable manifold that acts as a threshold for firing; (3) an up state having stability that depends on the time scale of the inhibition. In the case when the up state is stable, we look for wave fronts, which transition the media from a down to up state, and when the up state is unstable, we are interested in pulses, a transient increase in firing that returns to the down state. We explore the behavior of these waves as the time and space scales of the inhibitory population vary. Some interesting findings include bistability between a traveling front and pulse, fronts that join the down state to an oscillation or spatiotemporal pattern, and pulses which go through an oscillatory instability.
PubDate: 20180205T15:49:08Z
DOI: 10.1016/j.physd.2017.12.011
 Authors: Jeremy D. Harris; Bard Ermentrout
 Chaotic dynamics of largescale doublediffusive convection in a porous
medium Authors: Shutaro Kondo; Hiroshi Gotoda; Takaya Miyano; Isao T. Tokuda
Pages: 1  7
Abstract: Publication date: 1 February 2018
Source:Physica D: Nonlinear Phenomena, Volume 364
Author(s): Shutaro Kondo, Hiroshi Gotoda, Takaya Miyano, Isao T. Tokuda
We have studied chaotic dynamics of largescale doublediffusive convection of a viscoelastic fluid in a porous medium from the viewpoint of dynamical systems theory. A fifthorder nonlinear dynamical system modeling the doublediffusive convection is theoretically obtained by incorporating the Darcy–Brinkman equation into transport equations through a physical dimensionless parameter representing porosity. We clearly show that the chaotic convective motion becomes much more complicated with increasing porosity. The degree of dynamic instability during chaotic convective motion is quantified by two important measures: the network entropy of the degree distribution in the horizontal visibility graph and the Kaplan–Yorke dimension in terms of Lyapunov exponents. We also present an interesting on–off intermittent phenomenon in the probability distribution of time intervals exhibiting nearly complete synchronization.
PubDate: 20171227T13:08:29Z
DOI: 10.1016/j.physd.2017.08.011
Issue No: Vol. 364 (2017)
 Authors: Shutaro Kondo; Hiroshi Gotoda; Takaya Miyano; Isao T. Tokuda
 Phaselocking and bistability in neuronal networks with synaptic
depression Authors: Zeynep Akcay; Xinxian Huang; Farzan Nadim; Amitabha Bose
Pages: 8  21
Abstract: Publication date: 1 February 2018
Source:Physica D: Nonlinear Phenomena, Volume 364
Author(s): Zeynep Akcay, Xinxian Huang, Farzan Nadim, Amitabha Bose
We consider a recurrent network of two oscillatory neurons that are coupled with inhibitory synapses. We use the phase response curves of the neurons and the properties of shortterm synaptic depression to define Poincaré maps for the activity of the network. The fixed points of these maps correspond to phaselocked modes of the network. Using these maps, we analyze the conditions that allow shortterm synaptic depression to lead to the existence of bistable phaselocked, periodic solutions. We show that bistability arises when either the phase response curve of the neuron or the shortterm depression profile changes steeply enough. The results apply to any Type I oscillator and we illustrate our findings using the Quadratic IntegrateandFire and Morris–Lecar neuron models.
PubDate: 20171227T13:08:29Z
DOI: 10.1016/j.physd.2017.09.007
Issue No: Vol. 364 (2017)
 Authors: Zeynep Akcay; Xinxian Huang; Farzan Nadim; Amitabha Bose
 Resonance controlled transport in phase space
 Authors: Xavier Leoncini; Alexei Vasiliev; Anton Artemyev
Pages: 22  26
Abstract: Publication date: 1 February 2018
Source:Physica D: Nonlinear Phenomena, Volume 364
Author(s): Xavier Leoncini, Alexei Vasiliev, Anton Artemyev
We consider the mechanism of controlling particle transport in phase space by means of resonances in an adiabatic setting. Using a model problem describing nonlinear wave–particle interaction, we show that captures into resonances can be used to control transport in momentum space as well as in physical space. We design the model system to provide creation of a narrow peak in the distribution function, thus producing effective cooling of a subensemble of the particles.
Graphical abstract
PubDate: 20171227T13:08:29Z
DOI: 10.1016/j.physd.2017.09.010
Issue No: Vol. 364 (2017)
 Authors: Xavier Leoncini; Alexei Vasiliev; Anton Artemyev
 Initial–boundary value problems associated with the
Ablowitz–Ladik system Authors: Baoqiang Xia; A.S. Fokas
Pages: 27  61
Abstract: Publication date: 1 February 2018
Source:Physica D: Nonlinear Phenomena, Volume 364
Author(s): Baoqiang Xia, A.S. Fokas
We employ the Ablowitz–Ladik system as an illustrative example in order to demonstrate how to analyze initial–boundary value problems for integrable nonlinear differential–difference equations via the unified transform (Fokas method). In particular, we express the solutions of the integrable discrete nonlinear Schrödinger and integrable discrete modified Korteweg–de Vries equations in terms of the solutions of appropriate matrix Riemann–Hilbert problems. We also discuss in detail, for both the above discrete integrable equations, the associated global relations and the process of eliminating of the unknown boundary values.
PubDate: 20171227T13:08:29Z
DOI: 10.1016/j.physd.2017.10.004
Issue No: Vol. 364 (2017)
 Authors: Baoqiang Xia; A.S. Fokas
 Hyperbolic periodic orbits in nongradient systems and smallnoiseinduced
metastable transitions Authors: Molei Tao
Pages: 1  17
Abstract: Publication date: 15 January 2018
Source:Physica D: Nonlinear Phenomena, Volume 363
Author(s): Molei Tao
Small noise can induce rare transitions between metastable states, which can be characterized by Maximum Likelihood Paths (MLPs). Nongradient systems contrast gradient systems in that MLP does not have to cross the separatrix at a saddle point, but instead possibly at a point on a hyperbolic periodic orbit. A numerical approach for identifying such unstable periodic orbits is proposed based on String method. In a special class of nongradient systems (‘orthogonaltype’), there are provably local MLPs that cross such saddle point or hyperbolic periodic orbit, and the separatrix crossing location determines the associated local maximum of transition rate. In general cases, however, the separatrix crossing may not determine a unique local maximum of the rate, as we numerically observed a counterexample in a sheared 2Dspace Allen–Cahn SPDE. It is a reasonable conjecture that there are always local MLPs associated with each attractor on the separatrix, such as saddle point or hyperbolic periodic orbit; our numerical experiments did not disprove so.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.10.001
Issue No: Vol. 363 (2017)
 Authors: Molei Tao
 A computational exploration of the McCoy–Tracy–Wu solutions of the
third Painlevé equation Authors: Marco Fasondini; Bengt Fornberg; J.A.C. Weideman
Pages: 18  43
Abstract: Publication date: 15 January 2018
Source:Physica D: Nonlinear Phenomena, Volume 363
Author(s): Marco Fasondini, Bengt Fornberg, J.A.C. Weideman
The method recently developed by the authors for the computation of the multivalued Painlevé transcendents on their Riemann surfaces (Fasondini et al., 2017) is used to explore families of solutions to the third Painlevé equation that were identified by McCoy et al. (1977) and which contain a polefree sector. Limiting cases, in which the solutions are singular functions of the parameters, are also investigated and it is shown that a particular set of limiting solutions is expressible in terms of special functions. Solutions that are singlevalued, logarithmically (infinitely) branched and algebraically branched, with any number of distinct sheets, are encountered. The algebraically branched solutions have multiple polefree sectors on their Riemann surfaces that are accounted for by using asymptotic formulae and Bäcklund transformations.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.10.011
Issue No: Vol. 363 (2017)
 Authors: Marco Fasondini; Bengt Fornberg; J.A.C. Weideman
 Synchronization of impacting mechanical systems with a single constraint
 Authors: Michael Baumann; J.J. Benjamin Biemond; Remco I. Leine; Nathan van de Wouw
Pages: 9  23
Abstract: Publication date: 1 January 2018
Source:Physica D: Nonlinear Phenomena, Volume 362
Author(s): Michael Baumann, J.J. Benjamin Biemond, Remco I. Leine, Nathan van de Wouw
This paper addresses the synchronization problem of mechanical systems subjected to a single geometric unilateral constraint. The impacts of the individual systems, induced by the unilateral constraint, generally do not coincide even if the solutions are arbitrarily ‘close’ to each other. The mismatch in the impact time instants demands a careful choice of the distance function to allow for an intuitively correct comparison of the discontinuous solutions resulting from the impacts. We propose a distance function induced by the quotient metric, which is based on an equivalence relation using the impact map. The distance function obtained in this way is continuous in time when evaluated along jumping solutions. The property of maximal monotonicity, which is fulfilled by most commonly used impact laws, is used to significantly reduce the complexity of the distance function. Based on the simplified distance function, a Lyapunov function is constructed to investigate the synchronization problem for two identical onedimensional mechanical systems. Sufficient conditions for the uncoupled individual systems are provided under which local synchronization is guaranteed. Furthermore, we present an interaction law which ensures global synchronization, also in the presence of grazing trajectories and accumulation points (Zeno behavior). The results are illustrated using numerical examples of a 1DOF mechanical impact oscillator which serves as stepping stone in the direction of more general systems.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.10.002
Issue No: Vol. 362 (2017)
 Authors: Michael Baumann; J.J. Benjamin Biemond; Remco I. Leine; Nathan van de Wouw
 Inverse scattering transform and soliton solutions for square matrix
nonlinear Schrödinger equations with nonzero boundary conditions Authors: Barbara Prinari; Francesco Demontis; Sitai Li; Theodoros P. Horikis
Abstract: Publication date: Available online 19 December 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Barbara Prinari, Francesco Demontis, Sitai Li, Theodoros P. Horikis
The inverse scattering transform (IST) with nonzero boundary conditions at infinity is developed for an m × m matrix nonlinear Schrödingertype equation which, in the case m = 2 , has been proposed as a model to describe hyperfine spin F = 1 spinor Bose–Einstein condensates with either repulsive interatomic interactions and antiferromagnetic spinexchange interactions (selfdefocusing case), or attractive interatomic interactions and ferromagnetic spinexchange interactions (selffocusing case). The IST for this system was first presented by Ieda, Uchiyama and Wadati (2007) , using a different approach. In our formulation, both the direct and the inverse problems are posed in terms of a suitable uniformization variable which allows to develop the IST on the standard complex plane, instead of a twosheeted Riemann surface or the cut plane with discontinuities along the cuts. Analyticity of the scattering eigenfunctions and scattering data, symmetries, properties of the discrete spectrum, and asymptotics are derived. The inverse problem is posed as a RiemannHilbert problem for the eigenfunctions, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided. In addition, the general behavior of the soliton solutions is analyzed in details in the 2 × 2 selffocusing case, including some special solutions not previously discussed in the literature.
PubDate: 20171227T13:08:29Z
DOI: 10.1016/j.physd.2017.12.007
 Authors: Barbara Prinari; Francesco Demontis; Sitai Li; Theodoros P. Horikis
 PoissonNernstPlanck equations with steric effects  nonconvexity and
multiple stationary solutions Authors: Nir Gavish
Abstract: Publication date: Available online 16 December 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Nir Gavish
We study the existence and stability of stationary solutions of PoissonNernstPlanck equations with steric effects (PNPsteric equations) with two countercharged species. We show that within a range of parameters, steric effects give rise to multiple solutions of the corresponding stationary equation that are smooth. The PNPsteric equation, however, is found to be illposed at the parameter regime where multiple solutions arise. Following these findings, we introduce a novel PNPCahnHilliard model, show that it is wellposed and that it admits multiple stationary solutions that are smooth and stable. The various branches of stationary solutions and their stability are mapped utilizing bifurcation analysis and numerical continuation methods.
PubDate: 20171218T12:46:36Z
DOI: 10.1016/j.physd.2017.12.008
 Authors: Nir Gavish
 Monotonicity, oscillations and stability of a solution to a nonlinear
equation modelling the capillary rise Authors: Mateusz
Abstract: Publication date: 1 January 2018
Source:Physica D: Nonlinear Phenomena, Volume 362
Author(s): Łukasz Płociniczak, Mateusz Świtała
In this paper we analyse a singular secondorder nonlinear ODE which models the capillary rise of a fluid inside a tubular column. We prove global existence, uniqueness and find several approximations along with the asymptotic behaviour of the solution. Moreover, we are able to find a critical value of the nondimensional parameter for which the solution exhibits a transition in its behaviour: from being monotone to oscillatory. This is an analytical rigorous proof of the experimentally and numerically confirmed phenomenon.
PubDate: 20171212T12:38:45Z
 Authors: Mateusz
 Computing Evans functions numerically via boundaryvalue problems
 Authors: Blake Barker; Rose Nguyen; Björn Sandstede; Nathaniel Ventura; Colin Wahl
Abstract: Publication date: Available online 9 December 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Blake Barker, Rose Nguyen, Björn Sandstede, Nathaniel Ventura, Colin Wahl
The Evans function has been used extensively to study spectral stability of travellingwave solutions in spatially extended partial differential equations. To compute Evans functions numerically, several shooting methods have been developed. In this paper, an alternative scheme for the numerical computation of Evans functions is presented that relies on an appropriate boundaryvalue problem formulation. Convergence of the algorithm is proved, and several examples, including the computation of eigenvalues for a multidimensional problem, are given. The main advantage of the scheme proposed here compared with earlier methods is that the scheme is linear and scalable to large problems.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.12.002
 Authors: Blake Barker; Rose Nguyen; Björn Sandstede; Nathaniel Ventura; Colin Wahl
 Turing patterns in parabolic systems of conservation laws and numerically
observed stability of periodic waves Authors: Blake Barker; Soyeun Jung; Kevin Zumbrun
Abstract: Publication date: Available online 7 December 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Blake Barker, Soyeun Jung, Kevin Zumbrun
Turing patterns on unbounded domains have been widely studied in systems of reaction–diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the largeamplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from subcritical Turing bifurcations. This answers in the affirmative a question of OhZumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full smallamplitude stability diagram–specifically, determination of rigorous Eckhaustype stability conditions–remains an interesting open problem.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.12.003
 Authors: Blake Barker; Soyeun Jung; Kevin Zumbrun
 Travelling waves and their bifurcations in the Lorenz96 model
 Authors: Dirk L. van Kekem; Alef E. Sterk
Abstract: Publication date: Available online 5 December 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Dirk L. van Kekem, Alef E. Sterk
In this paper we study the dynamics of the monoscale Lorenz96 model using both analytical and numerical means. The bifurcations for positive forcing parameter F are investigated. The main analytical result is the existence of Hopf or HopfHopf bifurcations in any dimension n ≥ 4 . Exploiting the circulant structure of the Jacobian matrix enables us to reduce the first Lyapunov coefficient to an explicit formula from which it can be determined when the Hopf bifurcation is sub or supercritical. The first Hopf bifurcation for F > 0 is always supercritical and the periodic orbit born at this bifurcation has the physical interpretation of a travelling wave. Furthermore, by unfolding the codimension two HopfHopf bifurcation it is shown to act as an organising centre, explaining dynamics such as quasiperiodic attractors and multistability, which are observed in the original Lorenz96 model. Finally, the region of parameter values beyond the first Hopf bifurcation value is investigated numerically and routes to chaos are described using bifurcation diagrams and Lyapunov exponents. The observed routes to chaos are various but without clear pattern as n → ∞ .
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.11.008
 Authors: Dirk L. van Kekem; Alef E. Sterk
 New integrable model of propagation of the fewcycle pulses in an
anisotropic microdispersed medium Authors: S.V. Sazonov; N.V. Ustinov
Abstract: Publication date: Available online 2 December 2017
Source:Physica D: Nonlinear Phenomena
Author(s): S.V. Sazonov, N.V. Ustinov
We investigate the propagation of the fewcycle electromagnetic pulses in the anisotropic microdispersed medium. The effects of the anisotropy and spatial dispersion of the medium are created by the two sorts of the twolevel atoms. The system of the material equations describing an evolution of the states of the atoms and the wave equations for the ordinary and extraordinary components of the pulses is derived. By applying the approximation of the sudden excitation to exclude the material variables, we reduce this system to the single nonlinear wave equation that generalizes the modified sine–Gordon equation and the Rabelo–Fokas equation. It is shown that this equation is integrable by means of the inverse scattering transformation method if an additional restriction on the parameters is imposed. The multisoliton solutions of this integrable generalization are constructed and investigated.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.11.012
 Authors: S.V. Sazonov; N.V. Ustinov
 Decay of Kadomtsev–Petviashvili lumps in dissipative media
 Authors: S. Clarke; K. Gorshkov; R. Grimshaw; Y. Stepanyants
Abstract: Publication date: Available online 2 December 2017
Source:Physica D: Nonlinear Phenomena
Author(s): S. Clarke, K. Gorshkov, R. Grimshaw, Y. Stepanyants
The decay of Kadomtsev–Petviashvili lumps is considered for a few typical dissipations –Rayleigh dissipation, Reynolds dissipation, Landau damping, Chezy bottom friction, viscous dissipation in the laminar boundary layer, and radiative losses caused by largescale dispersion. It is shown that the straightline motion of lumps is unstable under the influence of dissipation. The lump trajectories are calculated for two most typical models of dissipation –the Rayleigh and Reynolds dissipations. A comparison of analytical results obtained within the framework of asymptotic theory with the direct numerical calculations of the Kadomtsev–Petviashvili equation is presented. Good agreement between the theoretical and numerical results is obtained.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.11.009
 Authors: S. Clarke; K. Gorshkov; R. Grimshaw; Y. Stepanyants
 Interaction of nonradially symmetric camphor particles
 Authors: ShinIchiro Ei; Hiroyuki Kitahata; Yuki Koyano; Masaharu Nagayama
Abstract: Publication date: Available online 2 December 2017
Source:Physica D: Nonlinear Phenomena
Author(s): ShinIchiro Ei, Hiroyuki Kitahata, Yuki Koyano, Masaharu Nagayama
In this study, the interaction between two nonradially symmetric camphor particles is theoretically investigated and the equation describing the motion is derived as an ordinary differential system for the locations and the rotations. In particular, slightly modified nonradially symmetric cases from radial symmetry are extensively investigated and explicit motions are obtained. For example, it is theoretically shown that elliptically deformed camphor particles interact so as to be parallel with major axes. Such predicted motions are also checked by real experiments and numerical simulations.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.11.004
 Authors: ShinIchiro Ei; Hiroyuki Kitahata; Yuki Koyano; Masaharu Nagayama
 Modeling ultrashort electromagnetic pulses with a generalized
Kadomtsev–Petviashvili equation Authors: A. Hofstrand; J.V. Moloney
Abstract: Publication date: Available online 2 December 2017
Source:Physica D: Nonlinear Phenomena
Author(s): A. Hofstrand, J.V. Moloney
In this paper we derive a properly scaled model for the nonlinear propagation of intense, ultrashort, midinfrared electromagnetic pulses (10100 femtoseconds) through an arbitrary dispersive medium. The derivation results in a generalized Kadomtsev–Petviashvili (gKP) equation. In contrast to envelopebased models such as the Nonlinear Schrödinger (NLS) equation, the gKP equation describes the dynamics of the field’s actual carrier wave. It is important to resolve these dynamics when modeling ultrashort pulses. We proceed by giving an orginal proof of sufficient conditions on the initial pulse for a singularity to form in the field after a finite propagation distance. The model is then numerically simulated in 2D using a spectralsolver with initial data and physical parameters highlighting our theoretical results.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.11.010
 Authors: A. Hofstrand; J.V. Moloney
 Wave propagation in a strongly nonlinear locally resonant granular crystal
 Authors: K. Vorotnikov; Y. Starosvetsky; G. Theocharis; P.G. Kevrekidis
Abstract: Publication date: Available online 23 November 2017
Source:Physica D: Nonlinear Phenomena
Author(s): K. Vorotnikov, Y. Starosvetsky, G. Theocharis, P.G. Kevrekidis
In this work, we study the wave propagation in a recently proposed acoustic structure, the locally resonant granular crystal. This structure is composed of a onedimensional granular crystal of hollow spherical particles in contact, containing linear resonators. The relevant model is presented and examined through a combination of analytical approximations (based on ODE and nonlinear map analysis) and of numerical results. The generic dynamics of the system involves a degradation of the wellknown traveling pulse of the standard Hertzian chain of elastic beads. Nevertheless, the present system is richer, in that as the primary pulse decays, secondary ones emerge and eventually interfere with it creating modulated wavetrains. Remarkably, upon suitable choices of parameters, this interference “distills” a weakly nonlocal solitary wave (a “nanopteron”). This motivates the consideration of such nonlinear structures through a separate Fourier space technique, whose results suggest the existence of such entities not only with a singleside tail, but also with periodic tails on both ends. These tails are found to oscillate with the intrinsic oscillation frequency of the outofphase motion between the outer hollow bead and its internal linear attachment.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.10.007
 Authors: K. Vorotnikov; Y. Starosvetsky; G. Theocharis; P.G. Kevrekidis
 Global dynamics for switching systems and their extensions by linear
differential equations Authors: Zane Huttinga; Bree Cummins; Tomáš Gedeon; Konstantin Mischaikow
Abstract: Publication date: Available online 15 November 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Zane Huttinga, Bree Cummins, Tomáš Gedeon, Konstantin Mischaikow
Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit thresholdlike behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an orderpreserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.11.003
 Authors: Zane Huttinga; Bree Cummins; Tomáš Gedeon; Konstantin Mischaikow
 On the manifestation of coexisting nontrivial equilibria leading to
potential well escapes in an inhomogeneous floating body Authors: Dane Sequeira; XueShe Wang; B.P. Mann
Abstract: Publication date: Available online 14 November 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Dane Sequeira, XueShe Wang, B.P. Mann
This paper examines the bifurcation and stability behavior of inhomogeneous floating bodies, specifically a rectangular prism with asymmetric mass distribution. A nonlinear model is developed to determine the stability of the upright and tilted equilibrium positions as a function of the vertical position of the center of mass within the prism. These equilibria positions are defined by an angle of rotation and a vertical position where rotational motion is restricted to a two dimensional plane. Numerical investigations are conducted using pathfollowing continuation methods to determine equilibria solutions and evaluate stability. Bifurcation diagrams and basins of attraction that illustrate the stability of the equilibrium positions as a function of the vertical position of the center of mass within the prism are generated. These results reveal complex stability behavior with many coexisting solutions. Static experiments are conducted to validate equilibria orientations against numerical predictions with results showing good agreement. Dynamic experiments that examine potential well hopping behavior in a waveflume for various wave conditions are also conducted.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.11.002
 Authors: Dane Sequeira; XueShe Wang; B.P. Mann
 Threewave resonant interactions: Multidarkdarkdark solitons,
breathers, rogue waves, and their interactions and dynamics Authors: Guoqiang Zhang; Zhenya Yan; XiaoYong Wen
Abstract: Publication date: Available online 10 November 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Guoqiang Zhang, Zhenya Yan, XiaoYong Wen
We investigate threewave resonant interactions through both the generalized Darboux transformation method and numerical simulations. Firstly, we derive a simple multidarkdarkdarksoliton formula through the generalized Darboux transformation. Secondly, we use the matrix analysis method to avoid the singularity of transformed potential functions and to find the general nonsingular breather solutions. Moreover, through a limit process, we deduce the general rogue wave solutions and give a classification by their dynamics including bright, dark, fourpetals, and twopeaks rogue waves. Ever since the coexistence of dark soliton and rogue wave in nonzero background, their interactions naturally become a quite appealing topic. Based on the N fold Darboux transformation, we can derive the explicit solutions to depict their interactions. Finally, by performing extensive numerical simulations we can predict whether these dark solitons and rogue waves are stable enough to propagate. These results can be available for several physical subjects such as fluid dynamics, nonlinear optics, solid state physics, and plasma physics.
PubDate: 20171212T12:38:45Z
DOI: 10.1016/j.physd.2017.11.001
 Authors: Guoqiang Zhang; Zhenya Yan; XiaoYong Wen
 Phase models and clustering in networks of oscillators with delayed
coupling Authors: Sue Ann Campbell; Zhen Wang
Abstract: Publication date: Available online 19 September 2017
Source:Physica D: Nonlinear Phenomena
Author(s): Sue Ann Campbell, Zhen Wang
We consider a general model for a network of oscillators with time delayed coupling where the coupling matrix is circulant. We use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to determine model independent existence and stability results for symmetric cluster solutions. Our results extend previous work to systems with time delay and a more general coupling matrix. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We apply our analytical results to a network of Morris Lecar neurons and compare these results with numerical continuation and simulation studies.
PubDate: 20170920T17:10:15Z
DOI: 10.1016/j.physd.2017.09.004
 Authors: Sue Ann Campbell; Zhen Wang